Atlas › Modular forms › Weight 4 › Level 176 › 176.4.a › 176.4.a.f

Newform orbit 176.4.a.f

$q$ -expansion

$q + 7q^{3} - 19q^{5} - 14q^{7} + 22q^{9} - 11q^{11} - 72q^{13} - 133q^{15} - 46q^{17} + 20q^{19} - 98q^{21} + 107q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ is a root of

$x + 1038$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 176$ )
11	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	7	−19	−14	−11	−72	−46	20	107	120	−117	−201	−228	242	96	458	−435	−668	−439	1113	−72	70	−358	895	409

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 176, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 4, 8, 11, 16, 22, 44, 88, 176]: [1, 2, 3, 5, 4, 8, 11, 21, 40, 76] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 6 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(176, 4)[5]   # 176.4.a.f

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 176:4 1000